Question

Integrate the function: \(\frac{xe^x}{(1+x)^2}\)

Answer 1

The correct answer is: \(∫\frac{xe^x}{(1+x)^2}dx=\frac{e^x}{1+x}+C\)
Let \(I=∫\frac{xe^x}{(1+x)^2} dx=∫e^x{\frac{x}{(1+x)^2}}dx\)
\(=∫e^x=[\frac{1+x-1}{(1+x)^2}]dx\)
\(=∫e^x=[\frac{1}{1+x}-\frac{1}{(1+x)^2}]dx\)
Let \(ƒ(x)=\frac{1}{1+x}\,\, ƒ'(x)=\frac{-1}{(1+x)^2}\)
\(⇒∫\frac{xe^x}{(1+x)^2}dx=∫e^x[{ƒ(x)+ƒ'(x)}]dx\)
It is known that,\(∫e^x[ƒ(x)+ƒ'(x)]dx=e^xƒ(x)+C\)
\(∴∫\frac{xe^x}{(1+x)^2}dx=\frac{e^x}{1+x}+C\)